If I fix b = d = y in [0,1] then I will get a statement (after a few lines)
for all epsilon > 0 there exists delta(depending on epsilon) > 0 such that, for all f_y in the set {f_y : y in [0,1]}, for all a,c in [0,1],
|(a,y) - (c,y)| < delta => | f_y (a) - f_y (c) | < epsilon

this is the definition for the set {f_y : y in [0,1]} to be uniformly equicontinuous.